How to save cut video as a video file in Blender 2.92.0
I found it! Turns out, it is the menu Render → Render Animation: it makes a window pop up, and after that window finishes playing, a video file appears.
But as I mentioned, I tried that menu and it didn't work, right? So why wasn't I finding it? Well, turns out when you set up the output directory for a file, you must end it with a slash. Otherwise the last /-separated chunk of the text would be interpreted as part of a resulting filename. Which means that if I wrote a path /tmp/foo (expecting the file to end up in the /tmp/foo directory that I created), then it saves to a file like /tmp/foo-SOME-POSTFIX.mkv, and I had too much stuff in my /tmp directory, so I simply missed it.